Control of Piecewise-Linear Hybrid Systems on Simplices and Rectangles

A necessary and sufficient condition for the reachability of a piecewise-linear hybrid system is formulated in terms of reachability of a finite-state discrete-event system and of a finite family of affine systems on a polyhedral set. As a subproblem, the reachability of an affine system on a polytope is considered, with the control objective of reaching a particular facet of the polytope. If the polytope is a simplex, necessary and sufficient conditions for the solvability of this problem by affine state feedback are described. If the polytope is a multi-dimensional rectangle, then a solution is obtained using continuous piecewise-affine state feedback.